I must say the mechanical time-counting mechanism is ingenious (see the photos in the original posting). Unfortunately, that's the root of the problem if the clock is losing time.
My best guess is that occasionally when the tens-minutes advances, it's somehow preventing the units-minutes from fully advancing.
Reminds me of a clock I made years ago with 10-position stepper-relays (it was originally a 5-digit voltmeter with incandescent bulbs). It "worked", and was very amusing to listen to it clicking-away at the top-of-the-hour when it had to advance the tens-minutes. 12-noon was quite a show.... But the relays were not sealed, so the contacts developed surface-corrosion and made intermittent contact, which caused the clock to lose time. After 20+ years of occasional usage I finally tossed it.